I've checked Kate's original version and the live (Austin) version. Plus that terrible cover version by EmmyLou.

And indeed the lyrics are different from what is given in the first post. In the chorus indeed there are 'tumbleweeds', also in the version of EmmyLou Harris.
Though the chorus is repeated a few times, hearing the 3 version I could not make out all the words. The best I could guess is that it goes something like:

It remains - blows down the dusty streets
And rides(?) the falling rain
And (g?)rows like a tumbleweed
Out on the open range(?)
The love I felt for you still remains

But I really do need help of any american or englishman who can make sense out of what is being sung. Is it range or plain etc, I can't make it out and Kate's poetry doesn't really make sense to me.

The verses inbetween are better to understand for me, they go like:

I went home to a place I swore I'd never see again
Lord it's different now there's hardly anything that hasn't changed
And the friends I've known have grown and moved away
But the love I felt for you still remains

I walked down past the building, standing empty and unused
Where you asked me one more time if I'd stay
And then you cried when I refused
But that was long ago, I moved away and I changed my name
But the love I felt for you still remains

I could say that I always thought you'd be there when I came
But you wanted more and I could say the same
Now I'm leaving like a whistle on a lonesome boxcar train
But the love I felt for you still remains

Note that there are a lot of differences with the lyrics earlier given in this thread. In some lines EmmyLou sings another line than Kate. I've written down Kate's words. Emmy Lou sings like a boxcar instead of whistle etc. But I'd like to forget her lamentable version, Harris version never should have been recorded...